Proceedings of the 8th International Conference on Sensing Technology, Sep. 2-4, 2014, Liverpool, UK
A New Method for Interference Reduction in the
Smoothed Pseudo Wigner-Ville Distribution
Stanislav Pikula
Petr Beneš
Brno University of Technology
Centre for Research and Utilization of Renewable Energy
Brno, Czech Republic
[email protected]
Brno University of Technology
Centre for Research and Utilization of Renewable Energy
Brno, Czech Republic
[email protected]
components in the signal result in outer interferences, whereas
a nonlinear FM signal produces inner interferences. Details
concerning the interferences and their formation are well
described by Hlawatsch and Flandrin [3].
Abstract— This article presents a new method facilitating
interference reduction in the Wigner-Ville distribution, which is
used for nonstationary signal analysis, for example in machine
condition monitoring. The algorithm is based on multiple
Smoothed Pseudo Wigner-Ville distributions: differently
smoothed time-frequency planes are compared and, for every
point, a cross-term free value is calculated on the basis of optimal
smooth estimation. The proposed approach is compared with the
Gabor-Wigner transform, the Zhao-Atlas-Marks distribution,
and the Choi-Williams distribution. Five time-frequency
Gaussian atoms and a bat echolocation chirp are used as the
testing signals. The Rényi entropy, the ratio of norms, the
Stanković measure, and the mean squared error are used as
quantitative measures to demonstrate the promising results of the
proposed method.
A. Interference reduction
The presence of cross terms diminishes the readability of
the result; thus, many methods for their reduction were
proposed, see Shafi et al. [4]. All such techniques attempt to
significantly reduce or completely remove the interferences,
preserve autoterm clarity (good resolution), and maintain the
maximum of the distribution properties. One of the basic
methods is the Pseudo WVD. Here, a filtering window is
introduced, whereby the result is filtered in the frequency
plane, and the interferences formed by distantly placed
components in the time plane are suppressed. However, cross
terms created by distant autoterms in the frequency plane
cannot be removed by frequency filtering.
Keywords- Wigner-Ville Distribution; Smoothed Pseudo
Wigner-Ville Distribution; Reduced Interference; Time-Frequency
Distribution; Quantitative measure
I.
For the removal of the cross terms formed in the frequency
plane, windowing in time is needed. In consequence of this
fact, smoothing in time and frequency was introduced. The
derived Smoothed Pseudo Wigner-Ville Distribution
(SPWVD) enables the reduction of all types of interferences at
the cost of resolution reduction, see Flandrin [5] or Fig. 1.
INTRODUCTION
As real life signals change over time (in, for example, the
analysis of heart sounds, machine condition monitoring, and
engine or gear fault diagnosis), transitional or nonstationary
components are an inevitable part of these signals. The
components can be hardly analyzed based on only the time or
the frequency, and therefore time-frequency tools were
introduced. Nevertheless, the widely used methods (such as the
Short Time Fourier Transform or the Wavelet Transform) do
not offer sufficient crisp resolution to facilitate detailed
examination
of
the
above-mentioned
components.
Consequently, a variety of Time-Frequency Distributions
(TFD) with different resolution levels and properties were
developed, see Boashash et al. [1].
Smoothing by wider time filtering (window width Lg) and
frequency filtering (window width Lh) leads to interference
reduction and slow degradation of the resolution, as is visible
in Fig. 1 (a)-(c) demonstrating the case of three Gaussian TF
atoms.
There is a number of other well-developed methods based
on the WVD; however, their details reach beyond the scope of
this article. The majority of effective algorithms are briefly
described and referenced in a review article by Shafi et al. [5].
In general, algorithms with good interference reduction are
iterative, signal-dependent, or parametric. The reduced
Interference Distributions (RIDs) derived from the general
TFDs constitute another class of solutions to the interference
issue, see Williams and Jeong [6].
The Wigner-Ville Distribution (WVD) is one of the basic
TFDs with optimal resolution, and it appears to be ideal for the
linear frequency modulated (FM) signal. In addition, the WVD
exhibits many beneficial mathematical properties, including
energy conservation, time and frequency shift invariants, or
compatibility with filters; the details are described by
Cohen [2].
II.
Good Time-Frequency (TF) resolution and other properties
are balanced by the presence of interferences. These constitute
an indivisible portion of the result if multi-component or
nonlinear FM signals are included in the analyzed signal. More
NEW METHOD
As mentioned above, the different time and frequency
window widths in the SPWVD affect the resulting presence of
the interferences and TF resolution. If we focus on the growing
window impact at different points in the TF plane, we can
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Proceedings of the 8th International Conference on Sensing Technology, Sep. 2-4, 2014, Liverpool, UK
Figure 1. Illustration of time and frequency smoothing on three TF Gaussian atoms and their three interferences: more intensive filtering reduces the
interferences and degrades the resolution. (a) The SPWVD of three TF Gaussian atoms, Lg=5 and Lh=252; (b) The SPWVD of the same signal, where the
interferences are reduced, Lg=9 and Lh=248; (c) The SPWVD of the same signal: interference-free and with a worse resolution, Lg=17 and Lh=240 .
distinguish three general cases. Firstly, the autoterm is affected
only a little in the beginning; however, with the growing
window width, its amplitude decreases, and the resolution
degrades; this process is shown in Fig. 2 (a). Secondly, the
interference amplitude rapidly decreases to zero – the
interferences are removed, see Fig. 2 (b). Lastly, if the
interference covers the autoterm, smoothing will rapidly
remove the interference, and the consecutive evolution matches
that of the autoterm, see Fig. 2 (c).
III.
VERIFICATION OF METHOD PERFORMANCE
A. Testing signals
For the TFD comparison, a set of simulated and real-life
signals is often used, see Shafi et al. [5]. The simulated signal
can be a number of TF Gaussian atoms (the number of atoms
ranges between one and five), a set of linear chirps, parabolic
or sinusoidal FM signals, or a combination of the abovementioned components. The applied real-life signals are either
a bat echolocation chirp or a whale echo, see Boashash et al.
[1] or Shafi et al. [5].
The proposed method needs to calculate a number of
SPWVDs with different filtrations. For a single TF point, our
algorithm finds the minimal difference between two
consecutive SPWVDs. Two SPWVDs with such minimal
difference then constitute the ideal smoothing for a concrete TF
point. The actual estimated value is calculated as the mean of
the two values provided by the two SPWVDs for the concrete
TF point. The described method is repeated for all TF points
within the whole TF plane.
The first chosen testing signal is the five TF Gaussian
atoms. The signal forms 10 outer interferences in the WVD.
The entire variety of outer cross terms is present, namely the
time, frequency, and TF interferences. In addition, one atom in
the center of the TF plane is masked by double interference,
and the majority of existing algorithms have problems with
autoterms overlaid by cross terms.
For the TF point corresponding to the autoterm, the first
difference (almost nonsmoothed) should be found as the best
estimation. While zero value should be chosen in the case of a
cross term, the first value with sufficiently reduced interference
is to be selected as the estimated one for the TF point
containing the interference and autoterm. The disadvantage of
the algorithm is the need of multiple calculations of the
SPWVD; however, this part of the algorithm can be
parallelized.
The second chosen testing signal is a brown bat
echolocation chirp. The signal was selected because of its real
life origin, wide use for performance comparison, and
availability.
B. Compared algorithms
For comparison with other methods, we first selected the
Gabor transform (GT) as the initial tool. The GT is a short-time
Fourier transform with a Gaussian window; although the
procedure provides only poor resolution, it completely
eliminates interferences. The next choice, characterized
Figure 2. Evolution of TF points with respect to growing filtering in time and frequency. (a) Evolution of the TF point corresponding to the autoterm; (b)
Evolution of the TF point corresponding to the interference; (c) Evolution of the TF point corresponding to the autoterm masked by the interference.
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Proceedings of the 8th International Conference on Sensing Technology, Sep. 2-4, 2014, Liverpool, UK
Figure 3. Comparison of results: (a) The proposed method based on 4 SPWVDs; the image shows degradation of the middle term, while the rest of the
interferences are visible; (b) The proposed method based on 16 SPWVDs: no degradation of the center term, and complete removal of the cross terms.
by simplicity and good properties, was a combination of Gabor
and the WVD: the Gabor-Wigner Transform (GWT), see Pei
and Ding [7].. Even though the GWT has four different
realizations, merely one of them is capable of reasonable
processing of the signal masked by interference (as in the case
of our first testing signal, namely the five TF atoms). The
corresponding GWT equation is

GWT  min{| GT (t , ) |2 , | WVD(t , ) |} 
and the middle component is slightly degraded; with the
growing number of used distributions, the estimation quality
nevertheless increases, see Fig. 3 (b). The results of the
quantitative measures applied to the two tested signals are
shown in Tabs. 1 and 2.

Finally, two representatives of the RIDs were chosen: the
cone-kernel representation, also named the Zhao-Atlas-Marks
distribution (ZAMD) by Zhao et al. [8], and the Choi-Wiliams
distribution (CWD) by Choi and Williams [9]. These two RIDs
are often compared with the SPWVD, see Hlawatsch et al.
[10]. Excepting the GWT, the reference algorithms were
realized using the GNU “The Time-Frequency Toolbox”.
TABLE I.
COMPARISON OF THE EXAMINED METHODS USING THE BAT
ECHOLOCATION; THE PROPOSED METHOD IS BASED ON 4 SPWVDS.
Used TFD
C. Quantitative measures
For quantitative comparison, more methods can be used.
Information theory offers the Shannon entropy; however,
Flandrin et al. [11] recommends the use of the Rényi entropy.
For our purpose, we applied the normalized Rényi entropy
presented by Sang and Williams [12]. Subsequently, the ratio
of norms (RN) by Jones and Parks [13] and the measure
introduced by Stanković [14] were considered. All these
techniques are commonly used to compare TFD algorithms.
Better distribution minimizes the Rényi entropy and
the Stanković measure. By contrast, the maximal RN value
corresponds to favorable distribution. For the simulated case of
the five TF atoms, we can calculate the ideal distribution result
as the sum of the five resulting WVDs of the mono-component
Gaussian atoms. It is therefore possible to use the mean
squared error (MSE) as the measure to compare the result
quality with the ideal WVD.
IV.
Out of all the tested techniques, the proposed method
(based on the four SPWVDs) provides the best performance for
the bat echolocation chirp. For the five Gaussian atoms, the
GWT produces slightly better results; however, as the masked
autoterm in the GWT incorporates oscillations, the MSE in the
proposed method performs better than the MSE applied in the
GWT.
Bat echolocation signal
Rényi
RN
Stanković
CWD
13.46
2.89
7.34
ZAMD
13.47
2.94
6.37
GT
14.90
0.73
8.99
GWT
12.98
3.09
3.73
Proposed
12.01
6.08
2.27
TABLE II.
COMPARISON OF THE EXAMINED METHODS USING THE FIVE
TF GAUSSIAN ATOMS; THE PROPOSED METHOD IS BASED ON 4 SPWVDS.
Used TFD
RESULTS
Even a small number of distributions is sufficient for
significant reduction of the discussed interferences in the
proposed method. In Fig. 3 (a), the result created on the basis
of the four SPWVDs is affected by remains of the cross terms,
Five time-frequency Gaussian atoms
Rényi
RN
Stanković
MSE
CWD
12.97
3.31
3.54
0.29
ZAMD
12.96
3.93
3.92
0.42
GT
13.84
1.18
3.28
0.45
GWT
11.84
4.68
0.92
0.04
Proposed
11.89
4.56
1.05
0.02
In its basic form, the designed algorithm is parametric.
The optimal choice of the window widths is generally a signaldependent parameter.
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Proceedings of the 8th International Conference on Sensing Technology, Sep. 2-4, 2014, Liverpool, UK
We obtained good results in selecting the window widths
depending on the number of the calculated SPWVDs (R) and
data length (N). The actual choice is expressed as follows:


 iN 
i {R,  ,1} : Fsmooth    
 2R 
 iN 
i {1,  , R} : Tsmooth    
 8R 
ACKNOWLEDGMENT


The author wishes to thank Curtis Condon, Ken White, and
Al Feng of the Beckman Institute of the University of Illinois
for the bat data and the permission to use them in this paper.
REFERENCES
In this form, the algorithm is nonparametric and exhibits
satisfactory results. The performance on the real-life bat
echolocation chirp is presented in Fig. 4 (a) and (b)., and the
result of the basic WVD is shown in Fig. 4 (b) for comparison.
V.
The authors gratefully acknowledge financial support from
the Ministry of Education, Youth and Sports of the Czech
Republic under projects No. LO1210 – “Energy for Sustainable
Development (EN-PUR)” solved at the Centre for Research
and Utilization of Renewable Energy.
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CONCLUSION
Even though a low number of SPWVDs is used, the
proposed algorithm shows robust results with a potential for
further enhancement based on a higher number of distributions.
In contrast to iterative algorithms, the algorithm applied in this
paper facilitates parallelization of its computationally most
intensive section (SPWVD calculations). The proposed method
showed the best result for both tested signals (the bat echolocation chirp and five time-frequency atoms) when compared
to the GWT, ZAMD, and CWD. In particular, the result of the
new method is notable for the case of the five time-frequency
atoms, since many existing interference reduction methods
have problems analyzing this type of signal. To conclude, the
proposed method is the one closest to the ideal WVD from all
the tested algorithms.
In the future, we plan to compare the above-outlined
technique with iterative algorithms, mainly because the method
offers the perspective of much faster computation and,
simultaneously, a resolution crisp close to the result expected
from the iterative algorithms. As a practical test, we plan to
carry out noise measurement of ventilator starts and stops in a
noised environment.
Figure 4. Comparison of the results:: (a) Result of the proposed method based on Bat echo: cross term free and with maximally preserved resolution; (b) The
Wigner-Ville distribution of the bat echo-location chirp (the result incorporates severe interferences).
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Proceedings of the 8th International Conference on Sensing Technology, Sep. 2-4, 2014, Liverpool, UK
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